L(s) = 1 | + 7-s + 1.41i·11-s − 13-s + 1.41i·17-s + 19-s − 1.41i·23-s + 1.41i·29-s + 31-s − 43-s − 1.41i·47-s − 1.41i·59-s + 61-s + 67-s + 1.41i·77-s + 1.41i·83-s + ⋯ |
L(s) = 1 | + 7-s + 1.41i·11-s − 13-s + 1.41i·17-s + 19-s − 1.41i·23-s + 1.41i·29-s + 31-s − 43-s − 1.41i·47-s − 1.41i·59-s + 61-s + 67-s + 1.41i·77-s + 1.41i·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.230051638\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.230051638\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + 1.41iT - T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + 1.41iT - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.726849853760162435945790524760, −8.571117989757483871185316019410, −8.058916373388292310969097090446, −7.15616098528516266801275248004, −6.55212368187699617938367334150, −5.15345747603836662233310407075, −4.83784542716465231509154655959, −3.80244267575712987763138962422, −2.43891429590937091663326239392, −1.56505875568933193688038560825,
1.04320301703188116681147743593, 2.50054492117105069112874598586, 3.37350855651480880371050693834, 4.62705114421934455860040065344, 5.27200937995422452314448589512, 6.05150573135710332496011069713, 7.25221067118758668698812393630, 7.77888556726321503794629606575, 8.516328304255330979798076045165, 9.481643109819209187920328009155